On the zeros of entire absolutely monotonic functions

By the definition, an entire absolutely monotonic function $f$ is an entire function representable in the form $f(z)=\int_0^{\infty}e^{zu}\,P(du)$, where $P$ is a nonnegative finite Borel measure on $\mathbf R^+$ and the integral converges absolutely for each $z\in\mathbf C$. This paper is devoted to the problem of characterization of the sets which can serve as zero sets of entire absolutely monotonic functions. We give the solution to the problem for the sets that do not intersect some angle $\{z:{|\arg z-\pi|}<\alpha\}$ for $\alpha>0$.